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f89a930 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 | import OAI.NumberTheory.Jacobsthal.Paths.GeometricRegularWords
namespace OAI
namespace Erdos970
open scoped _root_.Erdos970
namespace NumberTheoryLean.FirstIsolatedPosition
open PrimeHistories GeometricRegularWords IsolatedBinGeometry
open LogarithmicBinScale LogarithmicBinLabels
theorem first_matching_position {α : Type*} (P : α → Prop) (ps : List α)
(hex : ∃ p ∈ ps,P p) : ∃ pre p tail,ps=pre++p::tail ∧ P p ∧ ∀ q ∈ pre,¬P q := by
classical
induction ps with
| nil => simp at hex
| cons p ps ih =>
by_cases hp : P p
· exact ⟨[],p,ps,rfl,hp,by simp⟩
· have ht : ∃ q ∈ ps,P q := by
obtain ⟨q,hq,hP⟩ := hex
rcases List.mem_cons.mp hq with rfl | hq
· exact False.elim (hp hP)
· exact ⟨q,hq,hP⟩
obtain ⟨pre,q,tail,he,hq,hpre⟩ := ih ht
exact ⟨p::pre,q,tail,by rw [List.cons_append,he],hq,by simpa using And.intro hp hpre⟩
theorem geometric_first_isolated {w top xi C B R L alpha beta : ℝ}
(hw : 1 < w) (htop : w < top) (hxi : 0 < xi) (z : Node) (ps : List ℕ)
(hp : ps ∈ geometricWords hw htop hxi C B R L alpha beta z) :
∃ pre p tail,ps=pre++p::tail ∧
isolatedBin w top xi B alpha beta (label (zero_lt_one.trans hw) htop hxi p) ∧
∀ q ∈ pre,¬isolatedBin w top xi B alpha beta (label (zero_lt_one.trans hw) htop hxi q) :=
first_matching_position _ ps (geometric_isolated_prime hw htop hxi z ps hp)
end NumberTheoryLean.FirstIsolatedPosition
end Erdos970
end OAI
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